Q82 / 155 · WBJEE 2021 · Mathematics

Let $f: R \rightarrow R$ be such that $f(0)=0$ and $\left|f^{\prime}(x)\right| \leq 5$ for all $x$. Then $f(1)$ is in

  • A
    $(5,6)$
  • Correct answer
    B
    $[-5,5]$
  • C
    $(-\infty,-5) \cup(5, \infty)$
  • D
    $[-4,4]$

Solution

$\left|f^{\prime}(x)\right| \leq 5$ $\int_{0}^{1}-5 d x \leq \int_{0}^{1} f^{\prime}(x) d x \leq \int_{0}^{1} 5 d x$ $\Rightarrow-5 \leq f(1)-0 \leq 5$ $\Rightarrow-5 \leq f(1) \leq 5$ $\Rightarrow f(1) \in[-5,5]$

Hence, the correct answer is option: 'B'.

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